Definition Let SL(2,7) denote the group of all 2×2 matrices of determinant one over the finite field with 7 elements. Then G = PSL(2,7) is defined to be the quotient group SL(2,7) / {I,−I} obtained by identifying I and −I. In this article, we let G denote any group isomorphic to PSL(2,7).
G = PSL(2,7) has 168 elements. This can be seen by counting the possible columns; there are 72 − 1 = 48 possibilities for the first column, then 72 − 7 = 42 possibilities for the second column. We must divide by 7 − 1 = 6 to force the determinant equal to one, and then we must divide by 2 when we identify I and −I. The result is (48*42) / (6*2) = 168.
Simplicity of PSL(2,7)
It is a general result that PSL(n, q) is simple for n ≥ 2, q ≥ 2, unless (n, q) = (2,2) or (2,3). In the former case, PSL(n, q) is isomorphic to the symmetric group S3, and in the latter case PSL(n, q) is isomorphic to alternating group A4. In fact, PSL(2,7) is the second smallest nonabelian simple group, next to the alternating group A5 = PSL(2,5).
Actions on projective spaces
G = PSL(2,7) acts via linear fractional transformation on the projective line P1(7) over the field with 7 elements:
Every automorphism of P1(7) arises in this way, and so G = PSL(2,7) can be thought of geometrically as the group of symmetries of the projective line P1(7).
However, PSL(2,7) is also isomorphic to SL(3,2) (= GL(3,2)), the special (general) linear group of 3×3 matrices over the field with 2 elements. In a similar fashion, G = SL(3,2) acts on the projective Fano plane P2(2) over the field with 2 elements:
Again, every automorphism of P2(2) arises in this way, and so G = SL(3,2) can be thought of geometrically as the group of symmetries of the Fano plane.
Symmetries of the Klein Quartic
The Klein quartic x3y + y3z + z3x = 0 is a Riemann surface, the unique curve of genus 3 over the complex numbers C. The Klein quartic has automorphism group isomorphic to G. The order 168 of G is the maximum allowed for this genus. Klein's quartic pops up all over the place in mathematics, not least of which includes representation theory, homology theory, octonion multiplication, Fermat's Last Theorem, and Stark's theorem on imaginary quadratic number fields of class number 1!